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Astron. Astrophys. 333, 603-612 (1998)

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5. Slightly nonaxisymmetric mass loss

The angular momentum evolution of the star is altered dramatically if even a small amount of non-axisymmetry is allowed in the mass ejection process. If a shell is ejected aspherically, it generally carries a net momentum, and the direction of this momentum vector in general need not pass exactly through the center of mass. It is conceivable, for example, that the dust-formation instability found in spherically symmetric numerical simulations of the ejection process actually is non-axisymmetric, so that the forces exerted are not evenly distributed over the surface. In this way, the ejection process adds a small amount of angular momentum (`kick') to the star.

Suppose now that a large number of shells are ejected, adding small amounts of angular momentum in random directions. Since the simulations indicate that the shells are ejected with periods [FORMULA] of the order of the oscillation period of the star (on the order of a year) while the duration of the superwind phase is of the order of [FORMULA] yr, there are on the order of [FORMULA] kicks, each associated with an ejected mass [FORMULA] on the order of [FORMULA]. The maximum amount of angular momentum such a kick can impart is [FORMULA], where [FORMULA] is the ejection velocity, observed to be in the range 5-50 km/s. This maximum applies when the mass is ejected tangentially to the surface of the star. This is of course quite unrealistic, and one expects the angular momentum imparted to be only a small fraction of this:

[EQUATION]

where [FORMULA] is a small number. In the following, I estimate how large this number must be to explain the observed rotation periods.

The evolution of the star's angular momentum vector is obtained by adding the forcing by kicks to (13):

[EQUATION]

where [FORMULA] is a random fluctuating vector with time step [FORMULA] and amplitude [FORMULA]. This equation (Langevin's equation) is the same as that governing the Brownian motion of particles in a gas. Following the standard treatment in statistical physics (e.g. Becker, 1978) we can take the continuum limit, in which the time step is infinitesimal, and derive a Fokker-Planck equation for the probability distribution [FORMULA] of obtaining an angular momentum [FORMULA] after time t. Leaving out this derivation, the result is

[EQUATION]

where

[EQUATION]

is the `braking rate', [FORMULA] is the gradient in [FORMULA] -space, and D the diffusion coefficient in [FORMULA] -space

[EQUATION]

The main difference with respect to standard Brownian motion is that the coefficients [FORMULA] in the present case are functions of time.

If the kicks are random in direction, and the star initially non-rotating, the probability distribution f is isotropic in [FORMULA] -space, [FORMULA]. Writing

[EQUATION]

Eq. (20) can then be written as

[EQUATION]

If [FORMULA] and D are constant, as they are in the case of Brownian motion, the asymptotic solution [FORMULA] for large t is that for which the bracket on the RHS vanishes. This yields

[EQUATION]

i.e. a Maxwellian distribution peaking at [FORMULA]. In our case, [FORMULA] varies significantly with time, because the envelope mass varies strongly. We should therefore do not expect the distribution function to be a Maxwellian.

Before entering into more detailed calculations an estimate of the orders of magnitude to be expected can be made by making a quasi-stationary approximation to (20). The distribution F is then given approximately by (25). The typical angular momentum to be expected at the end of the mass loss, when the envelope mass left is [FORMULA] is then, with (21), (22):

[EQUATION]

where

[EQUATION]

is the number of kicks experienced. With (18):

[EQUATION]

where [FORMULA] for angular momentum loss at the photospheric value ([FORMULA] in 17). The expected rotation rate of the white dwarf, with gyration radius [FORMULA], mass [FORMULA] and radius [FORMULA] is then [FORMULA]. Assuming a final envelope mass of [FORMULA], initial envelope mass of [FORMULA], and [FORMULA], this yields

[EQUATION]

If shells are ejected every two years or so, we have [FORMULA]. A rotation period of 1d is then obtained for [FORMULA].

As long as it is not known how the relevant details of the ejection process take place, it is hard to argue whether an asymmetry of the order [FORMULA] is realistic or not, but a number as small as this would not seem too demanding. The reason why such small asymmetries are sufficient, even when their effect is further reduced by random superposition (the factor [FORMULA] in 28), is the very large lever arm on which the kicks act. A star on the AGB is so large compared with the final white dwarf that a very precisely axisymmetric mass loss would be needed to avoid introducing the small amount of angular momentum that is sufficient to produce white dwarfs with periods of a day.

5.1. Distribution of rotation rates resulting from random kicks

The coefficient [FORMULA] varies by a factor [FORMULA] as the envelope mass is reduced from its initial value to a representative post-AGB value of the order [FORMULA]. To take this into account, I solve Eq. (24) numerically. I use a second order , implicit time step and centered differences in the J -coordinate (Crank-Nicholson scheme).

As angular momentum coordinate I use the dimensionless variable j, defined by

[EQUATION]

Let

[EQUATION]

be the probability distribution per unit of [FORMULA], and for time coordinate use

[EQUATION]

Then Eq. (24) can be written as

[EQUATION]

The integration is from [FORMULA] to [FORMULA], where [FORMULA] is the envelope mass at which the mass loss ends. Apart from [FORMULA] the only parameter in the problem is the angular momentum loss index [FORMULA] (cf. Eq. 17). The evolution for [FORMULA] is given in Fig. 2, which shows the mean [FORMULA] of the probability distribution [FORMULA] as a function of the remaining envelope mass.

[FIGURE] Fig. 2. The mean angular momentum [FORMULA] as a function of the remaining envelope mass [FORMULA], for angular momentum induced by random non-axisymmetries in the superwind mass loss. Solid: solution of Eq. (33). Dotted: stationary approximation, Eq. (34). Evolution is from right to left

If the evolution is sufficiently `slow', one expects the solution to be close to the Maxwellian stationary solution, obtained by setting the square bracket in (33) equal to zero. This stationary distribution has mean dimensionless angular momentum

[EQUATION]

and is shown for comparison in Fig. 2. The stationary approximation actually turns out to be quite good, except in the initial phase of the evolution. The white dwarf rotation rate corresponding to (34) is given by:

[EQUATION]

Thus, the predicted white dwarf rotation rate decreases as the square root of the mass remaining in the envelope at the time when mass loss ceases.

5.2. Comparison with observed distribution

By adjusting either the asymmetry parameter [FORMULA] or the final envelope mass [FORMULA], the maximum of the distribution can be made to agree with the observations. This distribution is close to a Maxwellian, and its width is too narrow compared with the observations, which spread by a factor 20 or so. The factors influencing the mean rotation rate (35) most are the asymmetry parameter [FORMULA] and the remaining envelope mass [FORMULA]. Both might depend on systematic factors like the initial stellar mass. A random variable could be the phase in the thermal pulse cycle at which the superwind takes place, which is known to have an effect on the post-AGB evolution (Schönberner 1990, Vassiliadis & Wood 1993). Lacking a sufficiently detailed theory for the superwind, it is hard to guess how the asymmetry parameter might depend on such variables. Values of the remaining envelope mass, on the other hand, have been inferred for oscillating WD and post-AGB stars by comparisons with theoretical models. Clemens (1994) finds a hydrogen envelope masses of about [FORMULA]. In the helium (DB) white dwarfs and their possible progenitors the PG 1159 stars, only a helium envelope (with inferred masses of the order [FORMULA], cf. Dehner & Kawaler 1995) is left. Blöcker & Schönberner determine a hydrogen envelope mass of [FORMULA] for FG Sge. It seems reasonable to assume that a certain spread in [FORMULA] is present. This could be due, for example, to random variations in moment at which pulsation ceases. To fit the observed distribution with such a spread, I assume a log-normal distribution of the parameter [FORMULA], with peak at [FORMULA] and (1/e -) width from [FORMULA] to [FORMULA]. The asymmetry parameter is assumed to be [FORMULA]. The resulting period distribution is compared with the observations in Fig. 3. The agreement with the observations is not a test of the theory developed here (since both the width and the mean have been fitted), but comparison shows that a spread in envelope mass of a factor 5 on both sides of the mean is sufficient to explain the observed width.

[FIGURE] Fig. 3. Predicted distribution of rotation periods (solid) for asymmetry parameter [FORMULA], and a log-normal spread in final envelope mass from [FORMULA] to [FORMULA]. Histogram: observed distribution from Fig. 1

5.3. Binarity and common envelope

Alternatives to the picture sketched may be envisaged, in which the rotating white dwarfs are in one way or another the result of binarity. The following is a brief discussion of such scenarios.

The observed periods, [FORMULA] are suggestive of the orbital periods of close binaries, as has been noted by Schmidt et al. (1986), who suggest the possibility that AM Her stars might be the progenitors of the rotating white dwarfs. Those must then have somehow lost their companions, perhaps through the mass transfer. Current understanding of the evolution of CVs does not favor this possibility, since it predicts that the secondaries can not transfer all of their mass within a Hubble time. This is because the angular momentum loss slows down dramatically once the secondary has been reduced to a small degenerate dwarf (e.g. Verbunt 1996, Kolb 1993).

A second possibility that suggests itself is that of a binary companion absorbed in a common envelope (CE) process. Two different outcomes of such a CE are possible. One is that the envelope is ejected, by the orbital energy released, before the secondary has spiraled in completely. In the other, the secondary spirals in completely and merges with the primary. The first case leaves a detached system (such as V471 Tau) which then evolves into a CV by magnetic braking. Theory and numerical simulations (for reviews see Taam 1995, Livio 1996) predict that this case happens if the secondary is massive enough and the density gradient in the inner parts of the giant are not too steep. If these conditions are not met, the secondary is predicted to dissolve completely, transferring all its mass to the giant envelope. A significant fraction of common envelope systems may actually experience this fate.

The high incidence of elongated or bipolar structures in planetary nebulae (PN) and objects believed to be in transit from an AGB star to a PN suggests that a large fraction of PN involve some form of common envelope evolution (e.g. Han et al. 1995). Detailed hydrodynamical simulations have been made to reproduce the morphology of these nebulae (Icke et al. 1992, Frank & Mellema 1994). The results show that in the initial phases of the radiation driven nebular expansion there must have been a thick disk-like structure inhibiting fast outflow in the plane of the disk, leaving a structure of two rapidly expanding lobes and a more slowly expanding ring. In the CE interpretation, the disk contains the mass ejected in the spiral-in process.

If the secondary is small, the energy released as it spirals in is insufficient to eject the entire envelope of the primary. The net effect in this case is that both the mass and the angular momentum of the secondary are added to the envelope of the primary. The envelope remaining on the primary after the CE would then contain a large amount of angular momentum, even if the companion absorbed is small. Would this suffice to produce a rotating white dwarf? If our basic assumption of approximately uniform rotation is valid, the answer is negative. This follows from the analysis of Sect.  4.1, where I have shown that even a maximally rotating AGB star leaves a core rotating with a period of at least 10 years.

This answer applies as long as there is still a significant amount of mass left in the envelope after the common envelope process ([FORMULA], say), and mass loss then continues like in normal AGB stars. If any significant amount of mass is left in the form of a convective envelope after the CE, the results from Sect.  4.1predict that the result will be a very slowly rotating white dwarf.

The consequence of the above is that a rapidly rotating white dwarf by CE evolution is to be expected only if the final dissolution of the companion coincides rather precisely with the ejection of the last bits of envelope. Barring possible surprises conc erning late phases of CE evolution, the details of which are not well known, this situation would appear to be a rare coincidence.

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© European Southern Observatory (ESO) 1998

Online publication: April 20, 1998
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